Equilateral simplices in a four-dimensional normed space

Abstract

Let E be a four-dimensional real normed space, let P ⊂ E be a 3-plane, and let x ≥ 3/4 be a positive number. It is proved that there exist four equidistant points A1, A2, A3, A4 ∈ P and a point A5 ∈ E such that ‖A5Ai‖ = x · ‖A1A2‖ for is = 1, 2, 3, 4. In particular, E contains an equilateral simplex (with base in P). Bibliography: 5 titles.